CamSpec uses analytic approximations to the covariance matrices of the pseudo-spectra
derived under the assumptions of narrow window functions and uncorrelated (but anisotropic) noise pixel noise (\((\sigma ^T_i)^2\) ,
\((\sigma ^Q_i)^2\) , \((\sigma ^U_i)^2\) ) [1]}, [2]}, [3]}, [4]}.
The expressions are quite cumbersome, and so we give the
expresssions for only those covariance matrices used to form the TTTEEE likelihoods. The following equations
are based on the analytic formulae developed in [3]}, [4]}:
\(\langle \Delta \tilde{C}^{T_iT_j}_\ell \Delta \tilde{C}^{T_pT_q}_{\ell ^\prime } \rangle &\approx & ({\bar{C}}^{T_iT_p}_\ell {\bar{C}}^{T_iT_p}_{\ell ^\prime } {\bar{C}}^{T_jT_q}_{\ell } {\bar{C}}^{T_jT_q}_{\ell ^\prime })^{1/2} \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{(ip)(jq)} ) \nonumber \\& +& ({\bar{C}}^{T_iT_q}_\ell {\bar{C}}^{T_iT_q}_{\ell ^\prime } {\bar{C}}^{T_jT_p}_{\ell } {\bar{C}}^{T_jT_p}_{\ell ^\prime })^{1/2} \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{(iq)(jp)} ) \nonumber \\&+ & ({\bar{C}}^{T_jT_q}_\ell {\bar{C}}^{T_jT_q}_{\ell ^\prime })^{1/2} \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{2T(ip)(jq)}) \delta _{ip} \nonumber \\&+& ({\bar{C}}^{T_jT_p}_\ell {\bar{C}}^{T_jT_p}_{\ell ^\prime })^{1/2}\Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{2T(iq)(jp)} ) \delta _{iq} \nonumber \\&+ & ({\bar{C}}^{T_iT_p}_\ell {\bar{C}}^{T_iT_p}_{\ell ^\prime })^{1/2}\Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{2T(jq)(ip)}) \delta _{jq} \nonumber \\&+ & ({\bar{C}}^{T_iT_q}_\ell {\bar{C}}^{T_iT_q}_{\ell ^\prime })^{1/2} \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{2T(jp)(iq)}) \delta _{jp} \nonumber \\& + & \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{TT(ip)(jq)})\delta _{ip}\delta _{jq} +\Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{TT(iq)(jp)}) \delta _{iq}\delta _{jp},  \\\langle \Delta \tilde{C}^{T_iE_j}_\ell \Delta \tilde{C}^{T_pE_q}_{\ell ^\prime } \rangle &\approx & ( \bar{C}^{T_iT_p}_\ell \bar{C}^{T_iT_p}_{\ell ^\prime } \bar{C}^{E_jE_q}_\ell \bar{C}^{E_jE_q}_{\ell ^\prime })^{1/2} \Xi _{TE}(\ell , \ell ^\prime , \tilde{W}^{(ip)(jq)} ) \nonumber \\&+& \bar{C}^{T_iE_q}_\ell \bar{C}^{T_pE_j}_{\ell ^\prime } \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{(iq)(jp)} ) \nonumber \\& + & (\bar{C}^{T_iT_p}_\ell \bar{C}^{T_iT_p}_{\ell ^\prime })^{1/2} \Xi _{TE}(\ell , \ell ^\prime , \tilde{W}^{2P(ip)(jq)} )\delta _{jq} \nonumber \\& + &(\bar{C}^{E_jE_q}_\ell \bar{C}^{E_jE_q}_{\ell ^\prime })^{1/2} \Xi _{TE}(\ell , \ell ^\prime , \tilde{W}^{2T(jq)(ip)} )\delta _{ip} \nonumber \\&+ & \Xi _{TE}(\ell , \ell ^\prime , \tilde{W}^{TP(ip)(jq)})\delta _{ip}\delta _{jq},  \\\langle \Delta \tilde{C}^{E_iE_j}_\ell \Delta \tilde{C}^{E_pE_q}_{\ell ^\prime } \rangle &\approx & (\bar{C}^{E_iE_p}_\ell \bar{C}^{E_iE_p}_{\ell ^\prime } \bar{C}^{E_jE_q}_\ell \bar{C}^{E_jE_q}_{\ell ^\prime })^{1/2} \Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{(ip)(jq)}) \nonumber \\&+ & (\bar{C}^{E_iE_q}_\ell \bar{C}^{E_iE_q}_{\ell ^\prime } \bar{C}^{E_jE_p}_\ell \bar{C}^{E_jE_p}_{\ell ^\prime })^{1/2} \Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{(iq)(jp)} ) \nonumber \\& +& (\bar{C}^{E_jE_q}_\ell \bar{C}^{E_jE_q}_{\ell ^\prime })^{1/2} \Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{2P(ip)(jq)})\delta _{ip} \nonumber \\& +& (\bar{C}^{E_jE_p}_\ell \bar{C}^{E_jE_p}_{\ell ^\prime })^{1/2}\Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{2P(iq)(jp)})\delta _{iq} \nonumber \\& +& (\bar{C}^{E_iE_p}_\ell \bar{C}^{E_iE_p}_{\ell ^\prime })^{1/2}\Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{2P(jq)(ip)})\delta _{jq} \nonumber \\& +& (\bar{C}^{E_iE_q}_\ell \bar{C}^{E_iE_q}_{\ell ^\prime })^{1/2}\Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{2P(jp)(ip)})\delta _{jp} \nonumber \\&+& \Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{PP(ip)(jq)})\delta _{ip}\delta _{jq} + \Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{PP(iq)(jp)}) \delta _{iq}\delta _{jp}, \qquad  \\\langle \Delta \tilde{C}^{T_iT_j}_\ell \Delta \tilde{C}^{T_pE_{q}}_{\ell ^\prime } \rangle &\approx & {1 \over 2} (\bar{C}^{T_iT_p}_\ell \bar{C}^{T_iT_p}_{\ell ^\prime })^{1/2} (\bar{C}^{T_jE_q}_\ell + \bar{C}^{T_jE_q}_{\ell ^\prime }) \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{(ip)(jq)}) \nonumber \\&+& {1 \over 2} (\bar{C}^{T_jT_p}_\ell \bar{C}^{T_jT_p}_{\ell ^\prime })^{1/2} (\bar{C}^{T_iE_q}_\ell + \bar{C}^{T_iE_q}_{\ell ^\prime }) \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{(jp)(iq)} ) \nonumber \\& +& {1 \over 2}(\bar{C}^{T_jE_q}_\ell + \bar{C}^{T_jE_q}_{\ell ^\prime }) \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{2T(ip)(jq)}) \delta _{ip} \nonumber \\& +& {1 \over 2}(\bar{C}^{T_iE_q}_\ell + \bar{C}^{T_iE_q}_{\ell ^\prime }) \Xi _{TE}(\ell , \ell ^\prime , \tilde{W}^{2T(jp)(iq)}) \delta _{jp}, \\\langle \Delta \tilde{C}^{E_iE_j}_\ell \Delta \tilde{C}^{T_pE_q}_{\ell ^\prime } \rangle &\approx & {1 \over 2} (\bar{C}^{E_jE_q}_\ell \bar{C}^{E_jE_q}{\ell ^\prime })^{1/2} (C^{T_pE_i}_\ell + C^{T_pE_i}_{\ell ^\prime })\Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{(ip)(jq)} ) \nonumber \\&+ & {1 \over 2} (\bar{C}^{E_iE_q}_\ell \bar{C}^{E_iE_q}{\ell ^\prime })^{1/2} (\bar{C}^{T_pE_j}_\ell + \bar{C}^{T_pE_j}_{\ell ^\prime }) \Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{(iq)(jp)} ) \nonumber \\& +& {1 \over 2}(\bar{C}^{T_pE_i}_\ell + \bar{C}^{T_pE_i}_{\ell ^\prime }) \Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{2P(ip)(jq)})\delta _{jq} \nonumber \\& + & {1 \over 2}(\bar{C}^{T_pE_j}_\ell + \bar{C}^{T_pE_j}_{\ell ^\prime })\Xi _{EE}(\ell , \ell ^\prime , \tilde{W}^{2P(jp)(iq)})\delta _{iq}, \\\langle \Delta \tilde{C}^{T_iT_j}_\ell \Delta \tilde{C}^{E_pE_q}_{\ell ^\prime } \rangle &\approx & (\bar{C}^{T_iE_p}_\ell \bar{C}^{T_jE_q}_{\ell ^\prime }) \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{(ip)(jq)}) \nonumber \\&+& (\bar{C}^{T_iE_q}_\ell \bar{C}^{T_jE_p}_{\ell ^\prime }) \Xi _{TT}(\ell , \ell ^\prime , \tilde{W}^{(iq)(jp)} ), \) 
